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The problem provides the following information:

1. \( m \angle X = 30^\circ \)
2. \( m \angle YT = 50^\circ \)
3. \( m \angle WRZ = 90^\circ \)
4. The line \( XY \) is tangent to the circle at point \( T \).

We are asked to find:

- \( m \angle YZ \)
- \( m \angle WZ \)
- \( m \angle TW \)

### Step 1: Use the fact that \( XY \) is tangent at \( T \)
From the information given, the tangent property implies that \( \angle XTY = 90^\circ \) since the angle between the tangent \( XY \) and the radius \( RT \) is always \( 90^\circ \).

### Step 2: Triangle properties
We can also use the sum of angles in the triangle \( YTZ \). The sum of the interior angles of any triangle is always \( 180^\circ \).

In triangle \( YTZ \), we know:
- \( m \angle XTY = 90^\circ \) (from the tangent property)
- \( m \angle YT = 50^\circ \) (given)

So, we can find \( m \angle YZT \):
\[
m \angle YZT = 180^\circ - 90^\circ - 50^\circ = 40^\circ
\]

### Step 3: Right-angle triangle \( WRZ \)
Given that \( m \angle WRZ = 90^\circ \), triangle \( WRZ \) is a right triangle. We may need additional information about this triangle (such as side lengths or specific angle measures) to find the other requested angles precisely.

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At this stage, we have:

- \( m \angle YZ = 40^\circ \)
- For \( m \angle WZ \) and \( m \angle TW \), we will need additional relationships or information to finalize the solution.

Would you like to explore more details or continue with specific steps on how to solve for \( m \angle WZ \) and \( m \angle TW \)? 

### Related Questions:
1. How does the tangent property help in solving other circle problems?
2. Can you apply the Pythagorean theorem to this situation? How?
3. What are the properties of angles in a cyclic quadrilateral?
4. How would you calculate the length of the tangent from a point outside a circle?
5. Why is the sum of angles in any triangle always \( 180^\circ \)?

### Tip:
Always remember that the tangent to a circle forms a right angle with the radius drawn to the point of tangency!

Dado: m∠X = 30°, m∠YT = 50°, m∠WRZ = 90°, XY es tangente al círculo. Encuéntrense: m∠YZ, m∠WZ y m∠TW.

Learn how to solve a geometry problem involving a circle with a tangent line. Given angles and the tangent properties, we find angles YZ, WZ, and TW using circle theorems and triangle properties. Suitable for Grades 10-12.

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